Bike lanes and sidewalks solve most of the problem.
As to cyclists running through lights and ignoring stop signs, I plead guilty, sometimes. If there is a fair amount of traffic, I scrupulously obey the rules. But if there is no traffic, or the nearest car is down at the end of the next block, I sail through, for the same reason that pedestrians routinely jaywalk. People moving by muscle power tend to take the path of least resistance. This includes following the law of inertia, which means you try to keep moving, and the law of people-over-forty-not-wanting-to-jump-up-and-down-off-bicycle-seats-unnecessarily. This is an unwieldly formal name, so this principle is generally known as "Sphinx's axiom."
I once strolled through a campus with a university president, an otherwise bright guy, who started bemoaning the trail students had worn in the grass across an otherwise immaculate quadrangle. I was impertinent enough to laugh, and point out the doors of the buildings that the students were obviously traveling between. I told him that his only solution was to plant barriers, or give in and provide a landscaped path. People on foot are simply not going to take the long way around just because the President is trying to grow grass on the shortest route. I don't think he liked my answer, but I wasn't sure I really wanted the job for which I was interviewing, so I didn't much care.
You can’t fight idiots. My local university has learned and has paved the shortcuts